Wave Optics

At lower end
Tension, T? = 2g = 20 N (due to the 2 kg block)
Velocity, v? = √ (T? /μ) = √ (20/μ)
Wavelength, λ? = 6 cm

At upper end
Tension, T? = (2 kg + 6 kg)g = 8g = 80 N (due to the block and the rope)
Velocity, v? = √ (T? /μ) = √ (80/μ) = √4 * √ (20/μ) = 2v?

Since frequency (f) remains the same:
f = v? /λ? = v? /λ?
⇒ λ? = λ? * (v? /v? )
⇒ λ? = λ? * (2v? /v? ) = 2λ?
⇒ λ? = 2 * 6 cm = 12 cm

β = λD / (d? + a? sinωt)
β? - β? = λD/ (d? - a? ) - λD/ (d? + a? )
= λD [ (d? + a? ) - (d? - a? ) / (d? ² - a? ²) ]
= 2λDa? / (d? ² - a? ²)

$I=I_{0}co{s}^{2}30°$

$=I_0{\left(\frac{\sqrt[]{3}}{2}\right)}^2=\frac{3}{4}{I}_0$ 

3d = 0.6mm

D = 80cm

= 800mm

Path difference is given by

BP – Andhra Pradesh = Dx

$\frac{}{}$ $\Rightarrow \frac{dy}{D}=\left(2n+1\right)\frac{\lambda }{2}$   [for Dark fringe at P]

n = 0, for first dark fringe

$\frac{dy}{D}=\frac{\lambda }{2}$

$\lambda =\frac{2d}{D}y$

$$ $=\frac{2d}{D}*\frac{d}{2}[y=\frac{d}{2},Given$  first dark fringe is observed on the screen directly opposite to one of the slits]

$\lambda =\frac{2*0.6*0.6}{800*2}$

$\lambda =450mm$

$\frac{\text{Energy}}{\text{Volume}}=\frac{{\mathrm{M}\mathrm{L}}^2{\mathrm{T}}^{-2}}{{\mathrm{L}}^3}=\left({\mathrm{M}\mathrm{L}}^{-1}{\mathrm{T}}^{-2}\right)$

 The distance between two successive bright fringes is fringe width $\left(\beta \right)$ .

$\beta =\frac{\lambda D}{d}=\frac{589*{10}^{-9}*1.5}{0.15*{10}^{-3}}=5.9\mathrm{m}\mathrm{m}$

In primary rainbow, red colour is at top and violet is at bottom.

Intensity of secondary rainbow is less in comparison to primary rainbow.

$I_A=9l+4l+2\sqrt[]{9l*4l}cos0°$

$=13l+12l=25l$

$I_B=9l+4l+2\sqrt[]{9l*4l}cos\pi$

$=13l-12l=l$

$\therefore I_A-I_B=24l$

${\lambda }_1=560nm,$ fringe width, B = 7.2 mm

$\Rightarrow B_1=\frac{D{\lambda }_1}{d}=7.2mm$

For ${\lambda }_2\to B_2=\frac{D{\lambda }_2}{d}=8.1mm$ ${}_{}{}_{}\frac{{}_{}}{}$

$\frac{B_2}{B_1}=\frac{{\lambda }_2}{{\lambda }_1}=\frac{8.1}{7.2}\Rightarrow {\lambda }_2\frac{81}{72}{\lambda }_1$

$=\frac{9}{8}*560nm$

= 630 nm